On the conjecture about the nonexistence of rotation symmetric bent functions

In this paper, we describe a different approach to the proof of the nonexistence of homogeneous rotation symmetric bent functions. As a result, we obtain some new results which support the conjecture made in this journal, i.e., there are no homogeneous rotation symmetric bent functions of degree > 2. Also we characterize homogeneous degree 2 rotation symmetric bent functions by using GCD of polynomials. 1 Motivation Since the introduction in the seventies by Rothaus [1], bent functions have been intensively studied in the past three decades, and widely used in cryptography and error-correction coding due to their nice cryptographic and combinatoric properties. For example, the highest possible nonlinearity of bent functions can be used to resist the differential attack and the linear attack in symmetric cipher. Recently, homogeneous rotation symmetric (Abbr. RotS) Boolean functions have attracted attentions (see [2, 3, 4]) because of their highly desirable property, i.e. they can be evaluated efficiently by re-using evaluations from previous iterations. Consequently, when efficient evaluation of the function (for example, design of some cryptographic algorithm, such as MD4 and MD5) is essential, these functions can serve as a good option. It is natural to ask what kind of homogeneous RotS bent functions exist. In fact, homogeneous bent functions are of interest in literature [5, 6, 7, 8, 9, 10]. St˘ anic˘ a and Maitra [6, 7] studied RotS bent functions up to 10-variables. They enumerated all RotS bent functions in 8-variables. 4 · 3776 such functions of degree 2 were found. However, they couldn't find any homogeneous RotS bent functions of degree 3,4 and 5 in 10 variables. Thus they made the following conjecture. Conjecture 1.1 There are no homogeneous rotation symmetric bent functions of degree > 2. Let us summarize known results related to the above conjecture. Observing that bent functions are in fact Hadamard difference sets, Xia et al.[8] showed that there are no homogeneous bent functions of degree n in 2n variables for every n > 3. By using the relationship between the Fourier spectra of a Boolean function at partial points and the Fourier spectra of its sub-functions, Meng et al.[9] got a low bound of degree for homogeneous bent functions. From the view point of nonlinearity, St˘ anic˘ a [11] obtained the following nonexistence results (see Section 2 for the notation SANF of a Boolean function): Theorem 1.2 The following hold for a homogeneous RotS f of …